where the Hermitian operator \(H= \vec{\nabla }_t^2+k^2n^2(r)\) ; \(k=2\pi /\lambda \) , where \(\lambda \)  is the wavelength; \(\vec{\nabla }_t^2\)  is the transverse Laplacian; \(n^2(r)=n_1^2(1-2\Delta f(r))\) ; \(f(r)\)  is the index profile as a function of the radial cylindrical coordinate \(r\)  only, \(\Delta = (n_1^2-n_2^2)/(2n_1^2)\)  is the index profile height parameter, with \(n_1>n_2\)  [1]}; parameters \(l\)  and \(m\)  are respectively the topological charge (azimuthal mode number) and the radial mode number of the unperturbed OAM mode (denoted \(OAM_{l,m}\) ) with an amplitude
\(O_{l,m} (r,\theta ) =\frac{1}{\sqrt{N_{l,m}}}F_{l,m}(r) e^{il\theta },\) 
